Mobile Robotics 2 A Compact Course on Linear Algebra Giorgio Grisetti Kai Arras Gian Diego Tipaldi Cyrill Stachniss Wolfram Burgard SA -1 Vectors • Arrays of numbers • They represent a point in a n dimensional space 2 Vectors: Scalar Product • Scalar-Vector Product • Changes the length of the vector, but not its direction 3 Vectors: Sum • Sum of vectors (is commutative) • Can be visualized as “chaining” the vectors. 4 Vectors: Dot Product • Inner product of vectors (is a scalar) • If one of the two vectors has , the inner product returns the length of the projection of along the direction of •If the two vectors are orthogonal 5 Vectors: Linear (In)Dependence • A vector is linearly dependent from if • In other words if can be obtained by summing up the properly scaled. • If do not exist such that then is independent from 6 Vectors: Linear (In)Dependence • A vector is linearly dependent from if • In other words if can be obtained by summing up the properly scaled. • If do not exist such that then is independent from 7 Matrices • A matrix is written as a table of values • Can be used in many ways: 8 Matrices as Collections of Vectors • Column vectors 9 Matrices as Collections of Vectors • Row Vectors 10 Matrices Operations • Sum (commutative, associative) • Product (not commutative) • Inversion (square, full rank) • Transposition • Multiplication by a scalar • Multiplication by a vector 11 Matrix Vector Product • The i component of is the dot product . • The vector is linearly dependent from with coefficients . 12 Matrix Vector Product • If the column vectors represent a reference system, the product computes the global transformation of the vector according to 13 Matrix Vector Product • Each can be seen as a linear mixing coefficient that tells how contributes to . • Example: Jacobian of a multi- dimensional function 14 Matrix Matrix Product • Can be defined through • the dot product of row and column vectors • the linear combination of the columns of A scaled by the coefficients of the columns of B. 15 Matrix Matrix Product • If we consider the second interpretation we see that the columns of C are the projections of the columns of B through A. • All the interpretations made for the matrix vector product hold. 16 Linear Systems • Interpretations: • Find the coordinates x in the reference system of A such that b is the result of the transformation of Ax . • Many efficient solvers • Conjugate gradients • Sparse Cholesky Decomposition (if SPD) • … • The system may be over or under constrained. • One can obtain a reduced system (A’ b’ ) by considering the matrix (A b ) and suppressing all the rows which are linearly dependent. 17 Linear Systems • The system is over-constrained if the number of linearly independent columns (or rows) of A’ is greater than the dimension of b’ . • An over-constrained system does not admit a solution, however one may find a minimum norm solution by pseudo inversion 18 Linear Systems • The system is under-constrained if the number of linearly independent columns (or rows) of A’ is greater than the dimension of b’ . • An under-constrained admits infinite solutions. The degree of infinity is rank (A’ )-dim(b’). • The rank of a matrix is the maximum number of linearly independent rows or columns. 19 Matrix Inversion • If A is a square matrix of full rank, then there is a unique matrix B=A -1 such that the above equation holds. • The ith row of A is and the jth column of A-1 are: • orthogonal, if i=j • their scalar product is 1 , otherwise. • The ith column of A-1 can be found by solving the following system: This is the ith column of the identity matrix 20 Trace • Only defined for square matrices • Sum of the elements on the main diagonal, that is • It is a linear operator with the following properties • Additivity: • Homogeneity: • Pairwise commutative: • Trace is similarity invariant • Trace is transpose invariant 21 Rank • Maximum number of linearly independent rows (columns) • Dimension of the image of the transformation • When is we have • and the equality holds iff is the null matrix • • is injective iff • is surjective iff • if , is bijective and is invertible iff • Computation of the rank is done by • Perform Gaussian elimination on the matrix • Count the number of non-zero rows 22 Determinant • Only defined for square matrices • Remember? if and only if • For matrices: Let and , then • For matrices: 23 Determinant • For general matrices? Let be the submatrix obtained from by deleting the i-th row and the j-th column Rewrite determinant for matrices: 24 Determinant • For general matrices? Let be the (i,j) -cofactor, then This is called the cofactor expansion across the first row. 25 Determinant • Problem: Take a 25 x 25 matrix (which is considered small). The cofactor expansion method requires n! multiplications. For n = 25, this is 1.5 x 10^25 multiplications for which a today supercomputer would take 500,000 years . • There are much faster methods , namely using Gauss elimination to bring the matrix into triangular form Then: Because for triangular matrices (with being invertible), the determinant is the product of diagonal elements 26 Determinant: Properties • Row operations ( still a square matrix) • If results from by interchanging two rows, then • If results from by multiplying one row with a number , then • If results from by adding a multiple of one row to another row, then • Transpose : • Multiplication : • Does not apply to addition! 27 Determinant: Applications • Find the inverse using Cramer’s rule with being the adjugate of • Compute Eigenvalues Solve the characteristic polynomial • Area and Volume: ( is i-th row) 28 Orthogonal matrix • A matrix is orthogonal iff its column (row) vectors represent an orthonormal basis • As linear transformation, it is norm preserving, and acts as an isometry in Euclidean space (rotation, reflection) • Some properties: • The transpose is the inverse • Determinant has unity norm ( ± 1) 29 Rotational matrix • Important in robotics • 2D Rotations • 3D Rotations along the main axes • IMPORTANT: Rotations are not commutative 30 Matrices as Affine Transformations • A general and easy way to describe a 3D transformation is via matrices. Translation Vector Rotation Matrix • Homogeneous behavior in 2D and 3D • Takes naturally into account the non- commutativity of the transformations 31 Combining Transformations • A simple interpretation: chaining of transformations (represented as homogeneous matrices) • Matrix A represents the pose of a robot in the space • Matrix B represents the position of a sensor on the robot • The sensor perceives an object at a given location p, in its own frame [the sensor has no clue on where it is in the world] • Where is the object in the global frame? p 32 Combining Transformations • A simple interpretation: chaining of transformations (represented ad homogeneous matrices) • Matrix A represents the pose of a robot in the space • Matrix B represents the position of a sensor on the robot • The sensor perceives an object at a given location p, in its own frame [the sensor has no clue on where it is in the world] • Where is the object in the global frame? Bp gives me the pose of the object wrt the robot p B 33 Combining Transformations • A simple interpretation: chaining of transformations (represented ad homogeneous matrices) • Matrix A represents the pose of a robot in the space • Matrix B represents the position of a sensor on the robot • The sensor perceives an object at a given location p, in its own frame [the sensor has no clue on where it is in the world] • Where is the object in the global frame? Bp gives me the pose of the object wrt the robot p AB p gives me the pose of B the object wrt the world A 34 Symmetric matrix • A matrix is symmetric if , e.g. • A matrix is anti-symmetric if , e.g. • Every symmetric matrix: • can be diagonalizable , where is a diagonal matrix of eigenvalues and is an orthogonal matrix whose columns are the eigenvectors of • define a quadratic form 35 Positive definite matrix • The analogous of positive number • Definition • • Examples • • 36 Positive definite matrix • Properties • Invertible , with positive definite inverse • All eigenvalues > 0 • Trace is > 0 • For any spd , are positive definite • Cholesky decomposition • Partial ordering : iff • If , we have • If , then • • 37 Jacobian Matrix • It’s a non-square matrix in general • Suppose you have a vector-valued function • Let the gradient operator be the vector of (first-order) partial derivatives Then, the Jacobian matrix is defined as 38 Jacobian Matrix • It’s the orientation of the tangent plane to the vector- valued function at a given point • Generalizes the gradient of a scalar valued function • Heavily used for first-order error propagation → See later in the course 39 Quadratic Forms • Many important functions can be locally approximated with a quadratic form . • Often one is interested in finding the minimum (or maximum) of a quadratic form. 40 Quadratic Forms • How can we use the matrix properties to quickly compute a solution to this minimization problem? • At the minimum we have • By using the definition of matrix product we can compute f’ 41 Quadratic Forms • The minimum of is where its derivative is set to 0 • Thus we can solve the system • If the matrix is symmetric, the system becomes 42.